Week 2 - EM 221 Introduction to Optimization and Modeling...

EMÜ 221
Introduction to Optimization and
Modeling
Ceren Tuncer Şakar
2015-2016 Fall
Week 2

2
Feasible Region
The feasible region of an LP is the set of all points satisfying all the LP’s
constraints and sign restrictions.
Feasible Solution
Any solution that satisfies all constraints
Infeasible Solution
A solution that is not feasible is said to be infeasible
Optimal Solution
The feasible soluton that gives the best objective function value

Slack and Surplus Variables

A linear program in which all the variables are non-
negative and all the constraints are equalities is said
to be in standard form .

Standard form is attained by adding slack variables to
"less than or equal to" type constraints, and by
subtracting surplus variables from "greater than or
equal to" type constraints.

Slack and surplus variables represent the difference
between the left and right sides of the constraints.

Slack and surplus variables have objective function
coefficients equal to 0.
3

4
Max z = 3x
1
+ 2x
2
+ 0
s
1
+ 0s
2
+ 0s
3
s.t.
2 x
1
+ x
2
+
s
1
= 100
x
1
+ x
2
+ s
2
= 80
x
1
+s
3
= 40
x
1
,
x
2
, s
1
, s
2
, s
3
≥ 0
Standard Form

5
Graphical Solution to a 2-Variable LP

6
Graphical Solution to a 2-Variable LP
X1
X2
10 20 40 50 60 80
finishing constraint
carpentry constraint
demand constraint
z = 60
z = 100
z = 180
Feasible Region
G
A
B
C
D
E
F
H
From figure, we
see that the set
of points
satisfying the
Giapetto LP is
bounded by the
five sided
polygon
DGFEH. Any
point on or in
the interior of
this polygon (the
shade area) is
in the feasible
region.

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